Star-planet tidal interactions may result in the excitation of inertial waves in the convective region of stars. In low-mass stars, their dissipation plays a prominent role in the long-term orbital evolution of short-period planets. Turbulent convection can sustain differential rotation in their envelope, with an equatorial acceleration (as in the Sun) or deceleration, which can modify the waves propagation properties. We explore in this first paper the general propagation properties of free linear inertial waves in a differentially rotating homogeneous fluid inside a spherical shell. We assume that the angular velocity background flow depends on the latitudinal coordinate only, close to what is expected in the external convective envelope of low-mass stars. We use i) an analytical approach in the inviscid case to get the dispersion relation, from which we compute the characteristic trajectories along which energy propagates. This allows us to study the existence of attractor cycles and infer the different families of inertial modes; ii) high-resolution numerical calculations based on a spectral method for the viscous problem. We find that modes that propagate in the whole shell (D modes) behave the same way as with solid-body rotation. However, another family of inertial modes exists (DT modes), which can propagate only in a restricted part of the convective zone. Our study shows that they are less common than D modes and that the characteristic rays and shear layers often focus towards a wedge - or point-like attractor. More importantly, we find that for non-axisymmetric oscillation modes, shear layers may cross a corotation resonance with a local accumulation of kinetic energy. Their damping rate scales very differently from what we obtain for standard D modes and we show an example where it is independent of viscosity (Ekman number) in the astrophysical regime in which it is small.
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